Bifurcations in Hamiltonian Systems: Computing Singularities by Gröbner Bases

Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter (auth.)3540004033, 9783540004035

The authors consider applications of singularity theory and computer algebra to bifurcations of Hamiltonian dynamical systems. They restrict themselves to the case were the following simplification is possible. Near the equilibrium or (quasi-) periodic solution under consideration the linear part allows approximation by a normalized Hamiltonian system with a torus symmetry. It is assumed that reduction by this symmetry leads to a system with one degree of freedom. The volume focuses on two such reduction methods, the planar reduction (or polar coordinates) method and the reduction by the energy momentum mapping. The one-degree-of-freedom system then is tackled by singularity theory, where computer algebra, in particular, Gröbner basis techniques, are applied. The readership addressed consists of advanced graduate students and researchers in dynamical systems.

Table of contents :
1. Introduction….Pages 1-18
2. Method I: Planar reduction….Pages 21-44
3. Method II: The energy-momentum map….Pages 45-68
4. Birkhoff normalization….Pages 71-84
5. Singularity theory….Pages 85-96
6. Gröbner bases and Standard bases….Pages 97-132
7. Computing normalizing transformations….Pages 133-151
A. Appendix….Pages 153-158
References….Pages 159-165
Index….Pages 167-169